Test for the nonexistence of photons Sherman Karp

Naval Electronics Laboratory Center, Code 2400, San Diego, California 92152 (Received 18 June 1976)

Extending the concept of the classical addition of wave packets to that of a random ensemble, a hypothesis has been proposed that treats the classical field as such an ensemble. Computations of the resulting field envelope and short-term counting statistics were also made. As a consequence, a test was proposed that would essentially measure the "shot noise" inherent in such a model and differentiate it from the discreteness resulting from the photodetection process. In this paper the test is examined in terms of the resources and environment necessary to perform it.

BACKGROUND In quantum radiation theory it was originally felt that two photons could not interfere. 1 This hypothesis was first tested by Forrester, Gudmundsen, and Johnson2

in 1955 when they mixed two filtered incoherent sources in a microwave cavity. With the advent of the laser a few years later, it became common practice to mix or heterodyne optical sources. In reconciling quantum radiation theory with these experiments, Mandel and Wolf3 proposed that indeed the photons did not mix, merely the wave packets. Thus they suggested that the wave packets could add in a classical manner without the photons having to mix. Since physics often consists of one, two and many, it was natural to consider the classical addition of a multitude of wavepackets; first by this author, Gagliardi and Reed4 for the case of de­terministic phase, and again by this author5 for the case of purely random phase. In the latter case, the wave packets were assumed to take the form

where kT is the average number of wave packets in (0, T), 2F1 the hypergeometric function, Ln(x) the Laguerre polynomial,

2kT E is the power in each quadrature component of s(t) ,

and η is the quantum efficiency. When the first and second moments m1 and m2 of the

count density were examined the result was

where φ was uniformly distributed in (0, 2π), and k and x1 were the variables of a Poisson random process. An exact computation of the envelope distribution of the radiation process was made together with a derivation of the short-term counting statistics of a photodetected current where the radiation ensemble was considered completely as a classical field. The probability den­sity of the envelope took the form

and the short-term count density took the form

By recognizing that the first moment was proportional to the received energy, which must decrease inversely as the square of the distance from the source, it was argued that the statistic

could yield one of three results, depending upon what the true nature of the radiation was and how energy was transferred from the field to the detector. Thus if we were to examine this statistic for a true thermal source and indeed it were Gaussian, the result would be Bose-Einstein statistics and the statistic in Eq. (6) would be zero. If, on the other hand, the envelope density in Eq. (2) were the correct envelope density, then the statistic in Eq. (6) would yield the value

1421 J. Opt. Soc. Am., Vol. 66, No. 12, December 1976 Copyright © 1976 by the Optical Society of America 1421

if energy were transferred by photons, or the value

Ωabs is the solid angle subtended by the molecular ab­sorption cross section to the source and Ωsource the solid angle into which the source radiates if the transfer were a pure wave phenomenon. It is the purpose of this paper to outline an experiment wherein the value of this statistic can be determined unequivocally.

EXPERIMENT

The difficulty in performing the experiment is to differentiate between a pure Gaussian field where the statistic is zero and a quantized classical field where the outcome is Eq. (8). For the scenario leading to Eq. (7) the statistic is on the order of unity and does not pose a difficult test. If we examine Eq. (8) we see that the statistic has the value

(Ac, the cross-sectional area for absorption, ~ 1-10 A2, and F is the ƒ number), even when the field is both focused and diffraction limited, containing a single t ime-space mode. Consequently, it is advisable to use as short a wavelength as can be dealt with using optical lenses. This is on the order of 1000 Å, for which

Since the statistic defined in Eq. (6) is a polynomial in the moments, a conceptually simple experiment can be performed using the sample moments. These take the form

with the expected values being

The latter relationship implies that the sample moments a re consistent estimates of the true moments. Insert­ing the sample moments into Eq. (6) in place of the actual moments resul ts in a test statistic, ρ, 6 whose expected value is ρ and whose variance is

var

where

is the vth central moment. This test statistic is

(μ'2 is replaced by [n/(n- l)]μ'2 so that it is unbiased6). If m1 is selected to be approximately unity, then the var(p) is approximately 45/n, so that the standard deviation is approximately 6.7/n 1 / 2 . Thus if each sam­ple is taken from a cw incoherent source with a sam­pling time of T seconds, the power required in the source is

Also we see that

implies

Consequently, if we sample with a nanosecond aperture at a rate of 108 samples per second, the source is de­fined as approximately 10 nW and 1000 MHz bandwidth, and the time required to compute one point is

A serious attempt to perform this experiment would require that allowance be made for source instabilities, aliasing, and other anomalies. This can be accomplished by passing the light through a beamsplitter and per­forming two balanced simultaneous experiments. The first with the beam at F= 1 to yield Eq. (10) and the second with F=f so that the corresponding statistic ρ0

yields the value

By making ƒ greater than three, the new statistic ρ' = ρ-ρ0 = 4η(Ac /λ2)[1- (1/ƒ2)] would be source indepen­dent and free from all anomalies except the receiver dark currents . By adequate cooling, these can also be circumvented.

On a final note we point out that it is not necessary to perform the experiment consecutively in t ime. For example, we know that the test statistic p is Gaussian distributed for large n.6 The mean converges to (4ηAc /F2λ2) + 0(1/n) with a standard deviation equal to var 1 / 2 (ρ)~0( l /n 1 / 2 ) as already pointed out. After about 108 samples the expected value is accurate enough; it is just the standard deviation that is not. Thus by repeat­ing the experiment 105 t imes with 108 samples each time, and then averaging the outcomes, the same result would be obtained. In fact, by compiling the cumulative mean value one should be able to observe the conver­gence on a simple plot. There is also a singularity that potentially exists when m'1 = 0, for then ρ- ∞. However, the probability that m'1 = 0 is the probability that all the k1 = 0. For m1= 1, this is

If the storage of 10 samples is excessive, then the test statistic ρ can be recas t by using unbiased estima­tors for each of the moments. These are

1422 J. Opt. Soc. Am., Vol. 66, No. 12, December 1976 JOSA Letters 1422

DISCUSSION

so that

has numerator and denominator which a re individually unbiased.

EFFECT OF STABLE INSTRUMENTS

If the average power of the source can be set with precision, then a possibility exists wherein the mea­surement time can be reduced. Let us first assume that to first order all the moments can be approximated by those of the Bose-Einstein density. We then have

When an electromagnetic field is detected in a quan­tum device, a quantum noise term is added to the exist­ing variance of the field.7 This is due to the discrete nature of the quantum detector and should not automati­cally be associated with the discrete character which may already exist in the field. In fact, it is the hypoth­esis in Ref. 5, that the discrete nature of the field is more adequately characterized in a manner similar to shot noise and results from the accumulation of an ensemble of discrete wave packets. Furthermore, it is the contention in Ref. 5 that these two quantizing ef­fects, that of the field and that of the detection process , a re distinct and separable—the first generated at the source, the lat ter at the receiver . While the experi­ment proposed is exacting, it should be conclusive in the sense that a more detailed classical description which also is consistent with the discrete nature of the source is doubtful. In addition, any discrete classical description of the field will of necessity have a quanti­zation or shot-noise te rm. This will manifest itself in the variance of the field which in turn will show up in the statistics of the counting process . Consequently, even if the calculation in Ref. 5 is incorrect, the ex­periment proposed is correct so long as the premise is correct . That is, if the two measurements for ρ can be balanced to one par t in 106, the stated results should hold independent of the statistics of the source, since we would merely be observing the shot-noise term.

A positive result of the experiment would imply that (i) an EM field is constituted of an ensemble of wave packets described in Eq. (1), and (ii) these wave packets obey Maxwell's equation individually and hence p r e ­clude the existence of photons.

Since

we have finally

Now a minimum exists for m1=0. 19714. Thus if we can set our source to 10% m1= 0. 1 9 - 0. 21 for which var(ρ) = 20. 7 /n. This would reduce the necessary mea­surement time by a factor of 2 compared to m1=1.

1 P. A. M. Dirac, Quantum Mechanics, 3rd ed. (Oxford U. P . , London), p. 9.

2A. T. Forrester, R. A. Gudmundsen, and P. O. Johnson, "Photoelectric Mixing of Incoherent Light," Phys. Rev. 99, 1601 (1955).

3 L. Mandel and E. Wolf, "Coherence Properties of Optical Fields, "Rev. Mod. Phys. 37, 231 (1965).

4S. Karp, R. M. Galiardi, and I. S. Reed, "Radiation Models Using Discrete Radiator Ensembles," Proc. IEEE 56, 1704 (1968).

5S. Karp, "Statistical Properties of Ensembles of Classical Wave Packets," J. Opt. Soc. Am. 65, 421 (1975).

6H. Cramer, Mathematical Methods of Statistics (Princeton U. P . , Princeton, N.J . , 1954).

7R. M. Galiardi, and S. Karp, Optical Communications (Wiley Interscience, New York, 1976) p. 62.

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